VD is your biggest concern. That skinny little charging wire can only take so much. As Old Biscuit says, the fridge should be fine during transit time. It is not like it gets to ambient temperature minutes after power is removed from it.
Good move getting the pure sine wave inverter. The weakest link is the size of the battery bank. If you have a pair of group 27, group 31 or a single 4D deep cycle you should be fine for even a long day on the road using the inverter.
I have a single 4D case size Lifeline AGM battery. 215ah rated. 7 seasons so far. Hoping for a couple more no problems. I also have the Magnum 2000W pure sine wave inverter/charger. AGM charge rates are different from wet cell batterys. The Magnum charger is programmable to suit wet cell, AGM or gel cell batteries.